# identity with permutation matrix

I was told σ is a permutation of set {1, 2, ..., n}. And the permutation matrix is the n-by-n matrix A_σ whose ith column is the vector e_σ(i).

First asked to write down permutation matrices for all permutation in S_3, which I did. But then I am asked to prove A_(στ) = (A_σ)(A_τ), where this τ comes out of nowhere. I am confused what it refers to. Does it mean that I need to prove this based on another random set of permutation noted τ. How should I approach this?

• You are right. Both $\sigma, \tau$ are permutation of the set. – user99914 Oct 8 '15 at 23:14
• What this is asking you to do is to show that if $\sigma$ and $\tau$ are any two of the permulations of $S_3$, then $A_\sigma\tau = A_\sigma A_\tau$. – Paul Sinclair Oct 9 '15 at 0:37